a) Show that if $p$ is a prime number and $P$ is a $p$-subgroup of a finite group $G$, then $[G:P]=[N_G(P):P]$(mod p), where $N_G(P)$ denotes the normalizer of $P$ in $G$.

b) Assume that $H$ is a subgroup of a finite group $G$, and that $P$ is a Sylow $p$-group of $H$. Show that if $N_G(P) \subset H$ then $P$ is a Sylow p-subgroup of $G$.

I have forgotten too much group theory to make much progress on this. I try to look at the action of $P$ on the set of cosets, $G/P$, but I must be missing something because not much connects to anything.